"Physics 122B Electricity and Magnetism"
Physics 122B Electricity and Magnetism Lecture 9 (Knight: 28.1 to 28.5) Current and Resistance April 16, 2007 Martin Savage The Electron Current We start with some “thought experiments” on a simple system. We have a parallel plate capacitor that has been charged, e.g. with glass and plastic rods. Now we connect the plates with a wire. What happens? The plates quickly become neutral, and we say that the capacitor has been discharged. Further study shows that while the discharge is taking place, the wire gets warm, a light bulb can be made to glow, and a compass needle can be deflected. These are indicators of current flow in the wire. 11/22/2011 Physics 122B - Lecture 9 2 Charge Carriers and Inertia Does the discharge occur because positive charges are moving to the negative plate, or because negative charges are moving to the positive plate? We have asserted that the current in metals is caused by the flow of negative electrons. The first + + + direct evidence that this was the case was provided in - - - 1916 by the Tolman-Stewart experiment, which showed + + + - that negative charges “go to the bottom” of an - - - accelerated conductor. + + + - - - We model a metallic conductor as a rigid lattice of + + + - - positive charges pervaded by a “sea” of conduction electrons, 1 per atom, that move freely in the material. 11/22/2011 Physics 122B - Lecture 9 3 The Electron Current - In a metallic conductor in electrostatic equilibrium, the conduction electrons move - around quite rapidly, but there is no net - movement of charge. This can be changed by “pushing” on the sea of electrons with an electric field, thereby causing the entire sea of electrons to move in a particular direction, like a gas or liquid flowing through a pipe. We define the electron current i as the number of The net motion, the “drift speed” vd, is electrons Ne that pass through superposed on the random thermal motions a cross section of wire or other of the individual electrons, and it is very conductor in a time interval Dt. slow, typically around 10-4 m/s. In other words: Ne = i Dt. 11/22/2011 Physics 122B - Lecture 9 4 Current and Drift Velocity If the electrons have an average drift speed vd, then on the average in a time This table gives n interval Dt they would travel a distance Dx in the wire, for various metals. where Dx = vd Dt. If the wire has cross sectional area A and there are n electrons per unit volume in the wire, then the number of electrons moving through the cross sectional area in time Dt is Ne = n A Dx = n A vd Dt = i Dt . Therefore, i nAvd 11/22/2011 Physics 122B - Lecture 9 5 Question 1 Which wire has the largest electron current? 11/22/2011 Physics 122B - Lecture 9 6 Conservation of Current Question: An electron current iA flows to the light bulb, passing point A, where it delivers some energy and makes the bulb glow. How much electron current iB then passes point B? Answer: All of it! iA=iB. Reason: the electrons don’t have anywhere else to go. What goes to the bulb must return from the bulb. The bulb cannot “use up” the electrons. Plumber’s Analogy 1: If water flows into a constant diameter pipe at 2.0 m/s, it must flow out of the pipe at the same speed. It cannot “pile up” in the pipe. This principle is called “Conservation of Current”. 11/22/2011 Physics 122B - Lecture 9 7 A Puzzle We discharge a capacitor that has been given a charge of Q = 16 nC, using a copper wire that is 2 mm in diameter and has a length of L = 20 cm. Assume that the electron drift speed is vd = 10-4 m/s. How long does it take to discharge the capacitor? (Note that L/vd = 0.2m/10-4 m/s = 2000 s = 33.3 min.) Points to consider: 1. The wire is already full of electrons. 2. The wire contains about 5x1022 conduction electrons. 3. Q = 16 nC requires about 1011 electrons. 4. A length L’ of wire that holds 16 nC of conduction electrons is 4x10-13 m. 5. L’/vd = 4x10-9 s = 4 ns. That is roughly the discharge time. 11/22/2011 Physics 122B - Lecture 9 8 Creating a Current… Non-Static Situation Suppose you want to slide a book across a table. If you give it a quick push, it moves but slows due to friction as soon as you remove your hand, and its kinetic energy becomes heat. The only way to make the book move at a constant speed is to continue pushing it. The sea of conduction electrons is similar to the book. If you push the electrons, you create a current, but they are not moving in vacuum, and collisions with other electrons and atoms soon dissipates their kinetic energy as heat. The only way to maintain the current of electrons is to push them, using an electric field. An electron current is a non-equilibrium motion in an E field. 11/22/2011 Physics 122B - Lecture 9 9 Establishing the Electric Field in a Wire (1) The figure shows two metal wires attached to the plates of a parallel plate capacitor, with their ends close together but not touching. The wires are conductors, so some of the charge from the capacitor plates spreads out along the wires as surface charge. E=0 inside all conductors..plates and wires. Now we connect the wires. What happens? The surface electrons can move, and do so. In ~10-9 s the sea of electrons shifts slightly, and the surface charges are rearranged into a non-uniform distribution of charges, as shown. Surface charges near the + and – plates reflect these charges, but surface charges become near-neutral half- way along the wire 11/22/2011 Physics 122B - Lecture 9 10 Establishing the Electric Field in a Wire (2) The figure shows the region of the wire near the neutral midpoint. The surface charge rings become more positive to the left and more negative to the right. In Chapter 26, we found that a ring of charge makes an on-axis E field that: 1. Points away from a positive ring and toward a negative ring; 2. Is proportional to the net charge of the ring; 3. Decreases with distance from the ring. The non-uniform surface charge distribution creates an E field inside the wire. This pushes the electron current through the wire 11/22/2011 Physics 122B - Lecture 9 11 Example: The Surface Charge on a Current-Carrying Wire Two 2.0 mm diameter charged rings, with charges ±Q, are 2.0 mm apart. What value of Q causes the electric field at the midpoint to be E=0.010 N/C? 1 zQ E 5 103 N/C 4 0 z 2 R 2 3/ 2 4 0 z R 2 2 3/ 2 Q E 1.6 1018 C 10e z 11/22/2011 Physics 122B - Lecture 9 12 Turning the Corner 11/22/2011 Physics 122B - Lecture 9 13 Question 2 Which surface charge distribution will produce the largest electron current? 11/22/2011 Physics 122B - Lecture 9 14 A Model of Conduction (1) Suppose E 0 : K 1 mv 2 3 kT 2 2 gives v 105 m/s. However, v 0. Now turn on an E field. The straight-line trajectories become parabolic, and because of the curvature, the electrons begin to drift in the direction opposite E, i.e., “downhill”. ax=F/m=eE/m so vx=vix+ axDt = vix+ Dt eE/m This acceleration increases an electrons kinetic energy until the next collision, a “friction” that heats the wire….energy is imparted to the atoms of the lattice. 11/22/2011 Physics 122B - Lecture 9 15 A Model of Conduction (2) One Collision Time-History over many Collisions ___ ___ eE ___ eE ___ e vd vx vix Dt Dt E m m m ne A ( Collision is independent of E ) i nAvd E m 11/22/2011 Physics 122B - Lecture 9 16 Example: The Electron Current in a Copper Wire The mean time between collisions for electrons in room-temperature copper is = 2.5x10-14 s. What is the electron current in 2.0 mm diameter copper wire where the internal field strength is E = 0.010 N/C? e (1.60 10-19 C)(2.5 10 14 s)(0.010 N/C) vd E 4.4 105 m/s m (9.1110-31 kg) n 8.5 1028 m-3 ; A r 2 (1.0 103 m) 2 3.14 10-6 m 2 i nAvd 1.2 1019 electrons/s 11/22/2011 Physics 122B - Lecture 9 17 Batteries and Current 11/22/2011 Physics 122B - Lecture 9 18 Electrical Current dQ I , in the direction of E dt 1 ampere 1 A 1 coulomb per second = 1 C/s DQ I Dt DQ eNe I ei Dt Dt 11/22/2011 Physics 122B - Lecture 9 19 Example: The Current in a Copper Wire From the previous example, i nAvd 1.2 1019 electrons/s What is the current I? I ei (1.60 10-19 C)(1.2 1019 s -1 ) 1.9 A How much charge passes through the wire in an hour? D Q I Dt (1.9 A)(3600 s) 6840 C 11/22/2011 Physics 122B - Lecture 9 20 Current and Electrons 11/22/2011 Physics 122B - Lecture 9 21 The Current Density in a Wire I ei nevd A I J current density nevd A Example: A current of 1.0 A passes through a 1.0 mm diameter aluminum wire. What is the drift speed of the electrons in the wire? I I (1.0 A) J 2 1.3 106 A/m 2 A r (5.0 104 m) 2 J (1.3 106 A/m 2 ) vd 1.3 104 m/s ne (6.0 1028 m -3 )(1.60 10 19 C) 11/22/2011 Physics 122B - Lecture 9 22 Kirchhoff’s Junction Law I I in out Ii i 0; summed over all the currents to any "junction". 11/22/2011 Physics 122B - Lecture 9 23 End of Lecture 9 Before the next lecture, read Knight: 29.1 to 29.4. 11/22/2011 Physics 122B - Lecture 9 24